Density operator idempotent

The Dirac density operator for the reference state is idempotent ... 

When the idempotent density operator p is constructed from orbital solutions of the Hartree-Fock equations, (Ti - ) = 0, it satisfies the commutator equation... 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

Two other properties of the density operators follow from its definition (10.3). First, it is Hermitian, that is, p (Z) = p(Z). Second it is idempotent, that is, satisfies the property... 

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the... 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

Dirac defines an idempotent density operator p, such that the kernel is Y.i < ,(r) ,< (r/). The OEL equations are equivalent to... 

A < 1. Some of these coefficients can be close to 1 (strongly occupied NSO) and some close to 0 (weakly occupied NSO), but in any case the post-HF density matrix loses its property = p. Turning to the basis of natural orbitals (NOs) results in the appearance of two spin components of the density matrix, p° and p. For RHF, ROHF, and UHF cases, these spin components are idempotent, being projectors on the subspace of the full MO space spanned by the occupied density operator is the Hermitian positive-semidefinite operator with a spur equal to the number of electrons. At the same time, in general, this operator does not possess any other specific properties such as, for example, idempontency. After the convolution over the spin variables, the density operator breaks down into two components whose matrix representation in the basis set of atomic orbitals (AOs) has the form... 

Let us start with a short discussion about the kind of transformation we are seeking for. As described in the Chapter 2, the density matrix corresponding to a pure state is a projector, which satisfies the following properties (Chapter 3) p = p" and Tr(/o ) = 1. On the other hand, for a statistically mixed state, p p and Tr(p ) < 1. Now, let us look at a density operator that is obtained from a mixed state operator p by a unitary transformation, p = UpU. The question is whether this operator can or cannot be a pure state operator. The trace and idempotency properties for the transformed operator become ... 

It makes an input trial density operator p, which is nearly idempotent, into a new purified version p , which is more nearly idempotent. Li et al. used Pm and constructed the following variational energy functional for a noninteracting N-electron system with a Hamiltonian h (or for a tight-binding model with a nonself-consistent h) ... 

Let us look at how to translate the idempotency requirement of the density operator (which will be discussed in the next section more extensively) into a tensor equation as, for example. 

This set of projection weights can be used to project out solutions. Projection weights are previously called partition functions. They are different from projection operators. Projection ojierators are idempotent while projection weights are not. For the current purpose g ( r) will be chosen to be a positive function that decreases faster than the sum of the atomic densities within subsystem a. Hence the projection weights as defined by eq.(18) will be able to project out local properties near subsystem a. Section 4 of this chapter discusses particular choices for g ( r). The projected charge density and energy density are... 

The field Fs(r) depends only on the density p(r) and the idempotent density matrix y,(r,r ) through the field z(r [yj). The field. elr) does not appear in the expression for Fs(r) because there is no electron-interaction operator in the... 

It may be shown (McWeeny, 1956, 1960) that (6.2.2) and (6.2.22) provide necessary and sufficient conditions for a stationary value of the energy, and that these density matrix equations may be solved without reference to the individual occupied orbitals. The idempotency condition (6.2.22) is characteristic of a projection operator, an interpretation that is discussed in later sections. 

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